A year ago, I did a one-semester long course on Abstract Algebra at my university. When we started, I was excited, because I knew the material presented in this course would be somewhat different from the things I had learned previously (including subjects like calculus, linear algebra, logic, graph theory, et cetera). I also knew Abstract Algebra is a very broad subject, and many modern subjects are rooted in its matter, including Algebraic Geometry, Algebraic Topology, Algebraic Number Theory, and many more subjects.

I would be disappointed, however. The course was hard. I didn't mind that (I suspected it would). I didn't mind others being naturally better at it than me (as I'm not a particularly talented mathematician).

I do think that I could have learned the material better if it would be presented in a different matter, though. Every student received a little booklet, containing course notes. All material was in there. I felt that the material was presented in quite a dense manner. I like to see a lot of worked examples of new material I'm learning, but there weren't many in there. I guess the booklet covered in 135 pages about the same material the abstract algebra book by Dummit and Foote covers in about 223 pages. I guess a somewhat bigger "reference book", like Dummit and Foote's or Fraleigh's book, would serve me better.

Furthermore, we had to do a lot of exercises from the booklet. Each week, we had to hand in assignments, consisting of exercises in the booklet. Everything went pretty fast. Your work would be checked and graded, but the "ideal" solutions would not be shown on, for example, a webpage. I also found that disappointing, because I often learn a lot from worked solutions to exercises I do not (completely) understand.

In the end, I barely passed the course and decided that I would not follow more abstract algebra courses in the future. This was "Abstract Algebra I". Next year, there were optional courses called "Abstract Algebra II" and "Abstract Algebra III". These courses contain subject matter on Rings, Fields, Galois Theory, Ideals, and more. This meant that, along with about half of my classmates who also would not be attending these further courses, I wouldn't learn about these interesting subjects.

Sure, I can buy some "easy" introductory abstract algebra book (like Pinter's) and work my way through it, but I doubt it's the "real deal" and I will understand the material as well as my classmates who will follow "Abstract Algebra II" and "Abstract Algebra III".

I think this way, many talent is wasted. The abstract algebra courses feel a bit like a rat race, in which only the very best survive (until the end of algebra III). I don't think this is o.k. . It seems to me that the notions developed abstract algebra could aid you in the study of other subjects that may not be explicitely related to abstract algebra.

Questions:

Should I even bother trying to change the way abstract algebra is currently taught at my university? Do you think these complaints are legitimate? Or should I stop whining and accept abstract algebra just isn't for me, nor ever will be?

If you think these complaints are, at least to a certain extend, legitimate, I wonder how you think I should tell my university professors to do things differently.

Just to be clear: I think I won't benefit from the changes made in the way this subject is taught at our university (if any changes are to be made at all), since by then I probably will have already finished my bachelor's degree. So this question is intented for future students.

@RahulNarain My impression is that this question probably does not have one, definite answer. I suspect the answers given will be a collection of opinions. I always thought questions like this ought to be community-wiki-hammered.
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Max MullerMar 7 '13 at 21:15

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I would recommend trying another book and seeing if you like it more and if it suits your needs better. For example, try "A Book of Abstract Algebra": Second Edition by Charles C Pinter (great little book and awesome price). Then, I would look to "A First Course in Abstract Algebra", 7th Edition by John B. Fraleigh and then step it up to "Abstract Algebra", 3rd Edition by David S. Dummit and Richard M. Foote. Also look at the books in your library and see which ones float your boat. Complaining might make you feel better, but it will likely not solve the problem and could make it worse IMO.
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AmzotiMar 7 '13 at 21:18

4 Answers
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Several comments: I think that most "easy" abstract algebra books are well worth reading if you find an intermediate or advanced text too dense. I am not aware of any abstract algebra that is not "the real deal".

I wonder what the rest of your mathematics background is. My answer here would be affected if, say, this was the first "real" math course you ever took, or on the other hand if you had attended courses like real analysis or topology before this class.

If this is your first "real" class, there is the possibility you just haven't finished your habilitation phase moving from "calculus" type courses to "proof" type courses. For the first 12 years of school, US students are taught a lot of things in the mathematical curriculum which do not really represent mathematics properly. It is often a shock to adjust to the real thing. I would hope you might consider trying abstract type courses again, in this case :) Sometimes it takes getting used to. If this is the case, I don't know if you have the experience to say that the way things are taught should change.

If you have had "real" math classes before, then it's still possible you just don't have a very pedagogically oriented professor. It is often very hard to change the way one teaches, and several profs don't have the patience to do it. If you think your prof is pretty open, then it would be a fantastic idea to go talk to him about how the course went. He would probably be very happy to see you take such an interest in it. It might not result in an algebra revolution at your school, but it might make it a bit better for your peers!

I had run into the exact same situation when I took a course in Abstract Algebra 1: called Group Theory. I also had a "Real" math course, it was a very rigorous proof oriented linear algebra class. But I ended up dropping the group theory course. The problem was the prof taught graduate level courses for 10 years, then after 10 years was asked to teach an intro Group Theory course. He went extremely fast, (I dropped it right before the midterm), so fast that he finished the course by 2/3 of the duration of the course. For this prof, he thought this course was a very easy course.
The problem with Group theory is that there is a lot of material and it is very dense and inter-connects so much that it is overwhelming, its almost like learning a new language. ALSO I agree with orlandpm above when he says:
"They like to think that some people are not "cut out" to be mathematicians, and they are happy to sideline these people"
This the type of mentality I see a lot when one goes higher in mathematics! This amounts to a kind of elitism.
IF University Math Departments really want students to progress further in the more Abstract-Pure mathematics, it should change the courses. In my opinion they should take the content of the course and reduce it in half and go deeper with this, than to cover so much. Also the proofs in Abstract algebra are more difficult to visualize than in Linear Algebra. They should make this one course and create 2 course called : Group Theory 1 and Group Theory 2. In this way, they can increase revenue as well.
To me Group Theory is the course that most students encounter when trying to go further into Abstract math, and due to these kinds of experiences, the course in Group Theory is now kind of used to turn away many students from math.
It has been my experience, that Group Theory has been used to block people going further in university abstract math.

I sympathize with you. The people who teach advanced math courses are often very talented mathematicians, and (in my opinion) very talented mathematicians frequently have the following two misconceptions:

They believe that the most concise explanation is the most effective pedagogical tool

They like to think that some people are not "cut out" to be mathematicians, and they are happy to sideline these people

Math is all about finding elegant abstractions, and often the most rewarding thing to find at the end of a long and challenging mathematical study is a single theorem or equation that concisely answers all of the questions you set out to answer. However, the student who has never put in that work and suffered through the challenging questions could never appreciate the concise answer in the same way. In my humble opinion, an equation is rarely ever a great explanation.

To give you an example from my experience: my freshman physics professor started out the course by listing elementary particles on the board and describing the way they interact. This information was a very concise and profound description of the universe, but it meant nothing to someone unacquainted with the subject. I imagine you would feel the same way if your abstract algebra professor began the first day of class by writing the definition of a "group" on the board and then proceeding to prove elementary facts.

The notes your professor provided may have been both concise and precise, but at the same time have no pedagogical value. There are a number of ways you could have supplemented the notes: finding related books and working their exercises, asking your professor or classmates for help, etc. Whether you should be forced to put in this extra effort is another story. If your professor suffers from "flaw #2" above, its possible he thinks that those really cut out to study abstract mathematics will be able to handle his minimalistic and theoretical approach. Ultimately it is not his job to "weed out" the unworthy, or to challenge you -- his job is to get you to understand concepts you have never encountered before.

I encourage you to talk to your professor, or to your department, but it may be fruitless. You can still learn a ton of interesting things by reading on your own and doing problems. I have had a lot more enjoyment reading about math since I graduated than I did in the context of any of my courses. As for abstract algebra in particular, remember that all "abstract" ideas are grounded in concrete examples. We wouldn't care so much about groups if we didn't study symmetries in euclidean space. Likewise, we wouldn't care about abstract rings or fields if we didn't have the archetypal examples of integers and rational numbers, respectively. There is no formula for what kind of self study gives the "best appreciation" of mathematics, but hopefully if you pick some good standard resources and read them with curiosity and enthusiasm, you will be headed off in the right direction.

#1 is very intersting, and I think very indicative of some important learning strategies. From the mathematician's point of view, the best definition is the simplest (=most concise?). Still, some people learn better "from the ground up" with very detailed prototypical examples. The consequences are interesting: when I read math by physicists, I always feel like "good God, this is so complicated!. Then a mathematician comes along and says "well that's just an algebra acting by blah on this module," and then it's perfectly clear.
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rschwiebMar 7 '13 at 22:15

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I guess the moral of the story is this: when you get in the habit of simplicity (as mathematicians do), you are able to modularize concepts and combine them to the point that your statements become very compact. Granted, that process takes a lot of RAM for non-mathematicians. (Some) Physicists, on the other hand, seem to have no limit to the capacity to reguritate some absurdly complex construction without much knowledge of its details. In other words, they are completely comfortable with complexity. It's pretty amazing we use the same mathematical tools!
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rschwiebMar 7 '13 at 22:18

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I agree with you. The line "well that's just an algebra acting by blah on this module" is a great explanation... but the catch is you have to already have an appreciation of what an "algebra" is and what a "module" is, etc. When explaining math to a non-mathematician, you can't start with the jargon.
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orlandpmMar 7 '13 at 22:23

It happens to me too that I learn a lot from solutions to exercises, and when I get books, I always get "the good one" and one with solutions.

About what you were talking about, I think that even when you're given an "official" booklet, as they do in my university too, you should go to a library and learn and study from other books, those booklets are never enough, plus they're always very dense as they're most times supposed to be like summaries.

Your complaints though, seem totally legitimate to me, just tell these same things to the teachers you had and know their opinion. Ask if algebra II and III are the same way, and don't give up so soon. Tell them that solutions to exercises, at least after the deadline to give them, should be given to students, and tell them that more material, recommended books, etc, should be given besides that booklet.

I would also say that teachers should approach these subject the most intuitive way possible, as it's one of the most abstract ones, and it's hard to follow if classes are just: theorem -> proof -> theorem 2 -> proof 2... that's what books are for. I would recommend the teachers to try to give more intuitive insight of the things they deal with in the course. In my university, it's actually the teachers who usually ask for recomendations for the next year, to know what should be changed, or what could be done better in a different way.